SERDAR GÜNER and DANIEL DRUCKMAN
IDENTIFICATION OF A PRINCESS UNDER INCOMPLETE
INFORMATION: AN AMARNA STORY
ABSTRACT. This article presents four analyses of an interaction between the
middle-Bronze Age Pharaoh Nibmuarea and the Babylonian king KadashmanEnlil as described in the Amarna letters (Moran [1992] The Amarna Letters,
The Johns Hopkins Universiy Press, Baltimore, Maryland). Intent on denying
the Pharaoh his daughter in marriage, the Babylonian king was faced with the
choice of sending messengers who could (“dignitaries”) or could not identify
(“non-dignitaries”) his missing sister in the Pharaoh’s court. Intent on marrying
the king’s daughter, the Pharaoh was faced with the choice of showing the sister or
showing someone else. Based on the assumption of complete information (game
1), the analysis revealed a dominant-strategy equilibrium: Nibmuarea shows the
sister and Kadashman-Enlil sends non-dignitaries. Based on the assumption of
one-sided incomplete information (Pharaoh’s misperception; game 2), the analysis revealed that the Pharaoh had a dominant strategy of showing the sister
irrespective of whether the king is keen or reluctant to learn about his sister’s
fate. Based on the assumption of one-sided incomplete information (KadashmanEnlil’s misperception; game 3), the analysis revealed that if non-dignitaries are
sent, the Pharaoh prefers showing someone other than his sister. Based on the assumption of two-sided incomplete information (game 4), the Pharaoh finds it more
beneficial to present the sister irrespective of whether his intentions are genuine or
feigned. With incomplete information, it is difficult to judge the other’s intentions;
the cost of being caught cheating by not showing the sister to knowledgeable
messengers was quite high. These analyses highlight the strategic uncertainty that
characterized this Bronze-Age interaction.
KEY WORDS: Dominant-strategy equilibria, Judging intentions, Middle-Bronze
Age diplomacy, One and two-sided incomplete-information games, Strategic
uncertainty
The Amarna letters dating to 1400–1350 B.C.E constitute a vivid
demonstration of Bronze Age balance-of-power calculations and
interactions in the Middle East. During that time, the kingdoms of
Egypt, Hatti, Mitanni, and Babylon formed alliances, waged wars,
and resisted threats. The 317 Amarna letters (written in Akkadian,
the lingua franca during the bronze age) prove the existence of a
triangle of powerful kingdoms, namely Egypt, Hatti, and Mitanni
Theory and Decision 48: 383–407, 2000.
© 2000 Kluwer Academic Publishers. Printed in the Netherlands.
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SERDAR GÜNER AND DANIEL DRUCKMAN
orbited by Babylon and a rising Assyria (Moran, 1992; Redford,
1992; Roux, 1992).
Alliances in those days were offered on the basis of “brotherhood” established by demands and offers of princesses, that is,
dynastic marriages (Cohen, 1996). There is evidence in the Amarna
letters of such diplomatic marriages linking Mitanni and Babylonian kings to Pharaohs of Egypt, the hegemonic rulers of that time
(Moran, 1992). These rulers sent their daughters to Pharaohs to secure the mighty Egyptian alliance. Rarely would a Pharaoh send his
daughter to be another ruler’s bride or would an Egyptian queen
marry a foreign ruler.1 In the first letter of the Amarna collection,
EA1 (EA refers to “El Amarna”), the Pharaoh Nibmuarea (Amenophis III) replies to Kadashman-Enlil, the Babylonian king: “As for
your writing me in order to aggrandize yourself (and) to put oil on
the head of a girl, you for your part sent me one present. Are we to
laugh?”.2
This paper analyzes EA1 in an interactive setting where rulers
misperceive each other. We propose first a complete-information game, and, then, one-sided and two-sided incomplete information games in strategic form. The latter develops the consequences of reciprocal suspicions implied in the letter between the
sender, Pharaoh Nibmuarea, and the addressee, Kadashman-Enlil.
A complete-information game model of the interaction at an ordinal
level is presented by us elsewhere (Druckman and Güner, 2000).
In the game analyses to follow, we reason backward from known
outcomes described in the letter to the processes leading to those
outcomes. Each game is construed as a thought experiment in the
sense of an “if ... then” simulation of a possible path that would
have led to the decisions that were made: in this case, Nibmuarea’s
decision on whether or not to present the princess and KadashmanEnlil’s decision on whether or not to send messengers who were dignitaries. If we assume that the kings were acting strategically, then
these are plausible depictions of the decision-making processes;
however, it may not have been the actual processes about which
the letter provides little information. The insights gained from the
analyses illuminate the importance of assumptions made about the
other’s intentions under conditions of strategic uncertainty.
IDENTIFICATION OF A PRINCESS UNDER INCOMPLETE INFORMATION
385
The paper is organized into several parts. First, we describe the
interaction as depicted in the first letter. Second, we analyze the
interaction under the assumption of complete information. Third,
we develop two games of one-sided incomplete information, one
focusing on Nibmuarea’s misperception of Kadashman-Enlil and
the other on the latter’s misperception of the former ruler. Fourth,
we construct a two-sided information game where each ruler misperceives the other. The paper concludes with some implications
for understanding the motives and perceptions of these Bronze Age
rulers.
EL AMARNA LETTER 1 (EA1)
EA 1 focuses on the identification of a Babylonian princess at the
Egyptian court.3 The princess is Kadashman-Enlil’s sister given
as a wife to Nibmuarea by his father. The letter is Nibmuarea’s
reply to Kadashman-Enlil’s previous (unfound) letter that seemingly
accused the Pharaoh of not presenting the princess to the Babylonian messengers. By using his ignorance about his sister’s fate,
Kadashman-Enlil refuses to give his daughter to Nibmuarea as another wife. Nibmuarea writes: “I have just heard what you wrote me
about, saying, ‘Here you are asking for my daughter in marriage, but
my sister whom my father gave you was (already) there with you,
and no one has seen her (so as to know) if now she is alive or if she
is dead’ ” (Moran, 1992: 1).
Nibmuarea indicates to Kadashman-Enlil that the proper way of
identifying the princess is by sending dignitaries as messengers: “...
Did you, however, ever send here a dignitary of yours who knows
your sister, who could speak to her and identify her? Suppose he
spoke with her. The men whom you sent here are nobodies ... There
has been no one among them who knows her, who was an intimate of
your father, and who could identify her” (Moran, 1992: 1). He adds:
“... And as for your writing me ’You addressed my messengers as
your wives were standing gathered in your presence, saying: Here
is your mistress who stands before you. But my messengers did not
know her, (whether) it was my sister who was at your side’ ... and
still you say: ‘Who is to identify her?’ Why don’t you send me
a dignitary of yours who can tell you the truth, the well-being of
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your sister here...?” (Moran, 1992: 1). Nibmuarea is therefore well
aware that Kadashman-Enlil has the options of sending dignitaries
and non-dignitaries as messengers to identify his sister.
Thus, Kadashman-Enlil had two options, either sending dignitaries or sending non-dignitaries. Based on our interpretation of the
letter, we assume that he had these options in the eyes of Nibruarea:
The king could not know with certainty that his sister was alive
or dead. However, even if Kadashman-Enlil could be certain that
his sister was dead, the choice between sending dignitaries or nondignitaries would serve another purpose, namely denying a princess
to the Pharaoh. In the complete information game presented below,
his dominant choice of sending non-dignitaries could be interpreted
in this way. And, in the one-sided incomplete information game,
also presented below, a Reluctant Babylonian king is not interested
in the fate of his sister. On the other hand, if Nibruarea thought that
the king was certain in his belief about his sister’s death, he would
assume that Kadashman-Enlil only had the choice of sending nondignitaries, leading him to maximize his own utilities in light of the
king’s determined course of action.
As for Nibmuarea’s options, Kadashman-Enlil is aware that the
Pharaoh could well present the princess or some other woman
from the Court. We deduce this from the Pharaoh’s remarks about
Kadashman-Enlil’s account of the story. Nibmuarea writes: “And as
for your writing me: ‘Perhaps the one my messengers saw was the
daughter of some poor man, or of some Kaskean, or the daughter
of some Hanigalbatean, or someone from Ugarit.4 Who can believe
them? The one who was at your side ..., she did not open her mouth.
One cannot believe them at all.’ These are your words. But if your
sister were dead, what reason would there be for one’s concealing
her death and our presenting someone else?” (Moran, 1992: 1).
Kadashman-Enlil and Nibmuarea interact strategically: Kadashman-Enlil has to think about what Nibmuarea could choose and
Nibmuarea likewise has to think about Kadashman-Enlil’s choices.
Their separate decisions mutually affect them. Game theory is concerned with interactive decisions and could reveal strategic aspects
of this particular interaction. These choices are revealed through the
simultaneous choices of the players. In strategic form, the game
consists of each player taking decisions without knowledge of the
IDENTIFICATION OF A PRINCESS UNDER INCOMPLETE INFORMATION
387
other player’s decisions. In this game, the Pharaoh does not know
whether the messengers are dignitaries and the Babylonian king
does not know whether the princess will be presented. A different
game would be based on sequential choices. For example, the king
would move first by choosing the type of messengers. The Pharaoh
would then respond to that choice with his decision to present the
princess or someone else. This kind of alternation of moves is referred to as a signaling game. The king signals his intentions by
choosing the type of messengers in his move at the start of the game.
(See Guner, 1998, for an example of a signaling game.)
An essential assumption of most game-theory applications is that
players will make the same probability assessments if they have
access to the same information. Referred to as the common knowledge assumption by Kreps (1990), it is a basis for our modeling.
We assume that both Nibruarea and Kadashman-Enlil develop their
beliefs about each other’s intentions from the same body of information. Supporting this assumption is the evidence from the letters
indicating that the two rulers are well acquainted with each other.
Any attempt to relax this assumption would require sophisticated
mathematics outside the scope of this article.
We address the following questions in this article: If both rulers
are informed about their respective incentives, what rational decisions should they take? If they misperceive each other, should
they still take the same decisions as if they were in a completeinformation case? What game-theoretic advice can one provide for
the interaction under different information conditions? Answers to
these questions amount to the detection of strategic rationality at
work many millennia ago as encapsulated in a clay tablet.
A COMPLETE-INFORMATION GAME
The players are Nibmuarea and Kadashman-Enlil. Nibmuarea
can either show the princess (S) or show someone else (−S).
Kadashman-Enlil can either send dignitaries as messengers to Nibmuarea’s court (D) or send non-dignitaries (−D). These choices are
shown in Figure 1.
For convenience, we set players’ payoffs to zero when the identification outcome is reached, that is, when Nibmuarea presents
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SERDAR GÜNER AND DANIEL DRUCKMAN
Figure 1. Identification under complete information.
the princess and Kadashman-Enlil sends dignitaries. We define the
players’ payoffs for other outcomes by comparing them with those
obtained from identification. This comparison is done by indicating
the distance between the payoffs players get from the identification
and those they obtain from each of the other outcomes.
We assume that Kadashman-Enlil cannot deny a daughter to Nibmuarea when the princess is identified. Nibmuarea suffers no cost
of being caught presenting someone else to the dignitaries and can
ask and get Kadashman-Enlil’s daughter as his wife. The difference
between the identification (0) and Nibmuarea presenting someone
else to Kadashman-Enlil’s dignitaries is that the latter can deny
his daughter to Nibmuarea. So, Kadashman-Enlil obtains d, d >
0, the payoff of not sending his daughter to the Pharaoh. As for
Nibmuarea, he cannot get a daughter from the Babylonian king, so
obtains −d, and, in addition, he suffers the cost of being caught as
a cheater which we denote as −c. When non-dignitaries are sent
and the Princess is either shown or not, Nibmuarea suffers no cost
of being caught as a cheater but he is denied Kadashman-Enlil’s
daughter. Kadashman-Enlil does not care about his sister; his aim is
to deny a daughter to Nibmuarea and to get a bride from Egypt “to
aggrandize himself”. Thus, in the upper-right and the lower-right
cells of Figure 1, Nibmuarea obtains −d and Kadashman-Enlil d.
The strict opposition of interest indicated by d and −d is intended
to reflect Kadashman-Enlil’s desire to aggrandize himself. Our interpretation of this letter suggests such a motive: By daring to deny the
Pharaoh his daughter in marriage, the king is seen to assert his claim
of equal status. This motive can be regarded to trump his concern
for the welfare of his sister, as we note below. Further, the assumption of strict opposition of interest serves the goal of parsimony by
limiting the number of possible motives considered. The impacts
IDENTIFICATION OF A PRINCESS UNDER INCOMPLETE INFORMATION
389
of this assumption are reflected in all the equilibria we computed.
Thus, there are two preference parameters: the dynastic marriage
issue symbolized by d and the cost of cheating symbolized by c.
By showing the princess, Nibmuarea obtains 0 if KadashmanEnlil sends dignitaries and −d if non-dignitaries are sent. By not
showing the princess, Nibmuarea obtains −d −c if KadashmanEnlil sends dignitaries and −d if non-dignitaries are sent. Hence,
showing the princess is better than showing someone else if dignitaries are sent and gives him the same payoff as showing someone else
when the Babylonian king sends non-dignitaries. The strategy of
showing the princess is therefore a dominant strategy.5 KadashmanEnlil too has a dominant strategy: sending non-dignitaries. He gets a
greater payoff if non-dignitaries are sent when the princess is shown
(d > 0), and, if the princess is not shown, it gives an equal amount
of utility as sending dignitaries, that is d = d. Hence, both players
have dominant strategies in this game.
The game has a dominant-strategy equilibrium: Nibmuarea
shows the princess and Kadashman-Enlil sends non-dignitaries.
Thus, the equilibrium outcome is (S, −D). Both players choose
their dominant strategies even without reasoning that “if I choose
not to show the princess, then if he sends dignitaries this will be my
worst outcome”, or “if I send dignitaries but he does not show my
sister then I will catch him trying to deceive me”, and so on. Such
calculations due to strategic uncertainty simply do not exist, because
neither Nibmuarea nor Kadashman-Enlil can think that dominated
strategies will be chosen. Neither Nibmuarea nor Kadashman-Enlil
can expect the other to choose his dominated strategy since each
knows the other knows that each knows and ad infinitum that they
are rational. Thus, the complete-information game explains the observed outcome as a result of dominant strategies, given that both
rulers are aware of these options and that this is common knowledge.
ONE-SIDED INCOMPLETE-INFORMATION GAMES
The assumption of complete information is not realistic. Players
could not have been sure that each had a unique preference ordering.
We now relax this assumption. Accordingly, we propose two games
of one-sided incomplete-information to understand the connections
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SERDAR GÜNER AND DANIEL DRUCKMAN
Figure 2. Identification under Nibmuarea’s misperception.
that can exist between belief systems and the observed outcome. The
first game models Nibmuarea’s misperception about KadashmanEnlil and the second models Kadashman-Enlil’s misperception
about Nibmuarea. Each is an extension of the complete-information
game taking these suspicions into account.
Nibmuarea argues that the observed outcome is the presentation of the princess to the non-dignitaries, and, contrary to this,
Kadashman-Enlil pretends that someone other than his sister is
presented to the non-dignitary messengers sent by him. To develop
the incomplete-information game in the direction of such uncertainties, Nibmuarea’s and Kadashman-Enlil’s strategy choices are
assumed to become meaningful given the private information they
possess.
Pharaoh’s misperception
Nibmuarea could have misperceived Kadashman-Enlil’s intentions
in two ways: one who is keen on learning his sister’s fate and another
who is not. The former type is called the “Keen” and the latter the
“Reluctant”. The matrices shown in Figure 2 represent the game
where Nibmuarea is uncertain about Kadashman-Enlil’s preferences
but Kadashman-Enlil knows that the Pharaoh has no incentive to
present someone else to either dignitaries or non-dignitaries, as he
IDENTIFICATION OF A PRINCESS UNDER INCOMPLETE INFORMATION
391
affirms in EA1. The first game matrix represents Pharaoh’s perception that Kadashman-Enlil is the Keen on learning his sister’s fate.6
The second one represents Pharaoh’s perception that the Babylonian ruler is the Reluctant. If p = 0, the game reduces to the
complete-information game previously discussed.
The Keen suffers a cost of −l, l > 0 in all outcomes other than
the identification. This parameter measures the Keen’s suffering of
not learning whether his sister is well. It is assumed that d < l, that
is, for the Keen learning his sister’s fate represents a greater utility
than the denial of a daughter to the Pharaoh insisting on a bride
from the Egyptian Court. The Reluctant does not suffer that cost, in
that case l = 0. The Reluctant has the same preference ordering as in
the complete-information version. He cares only about aggrandizing
himself. Thus, the preference parameter l helps to distinguish the
two types of Kadashman-Enlil. It indicates the private information
that the Babylonian ruler possesses.
Nibmuarea’s prior assessments about Kadashman-Enlil’s type
are that he is Keen with probability p and Reluctant with (1 − p).
These probabilities are Pharaoh’s prior assessments of KadashmanEnlil. They are assumed to be common knowledge. Pharaoh’s
assessments are known by the Babylonian ruler; Pharaoh knows
that the Babylonian ruler is informed about his misperception about
him; or the Babylonian ruler knows that the Pharaoh knows that
the Babylonian ruler knows his misperception about him, and ad
infinitum. Kadashman-Enlil knows his own preference ordering.
Kadashman-Enlil’s payoffs in outcomes other than the identification are negative if he is the Keen: d − l < 0 as d < l. This
implies that the Keen has a dominant strategy of D and the Reluctant has the dominant strategy of −D as in the complete-information
game. This forms a type-contingent strategy for Kadashman-Enlil:
“if I am the Keen, I send dignitaries, but, if I am Reluctant, I send
non-dignitaries” denoted as (D, −D). The other type-contingent
strategies (D, D), (−D, D), (−D, −D) are ruled out (the first letter
refers to Keen’s choice and the second one to that of the Reluctant) as none of the types should choose his dominated strategy.
Nibmuarea has only one type, so he only has the choice of either
showing the princess (S) or showing someone else (−S). He has his
dominant strategy of S as in the game of complete information.
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SERDAR GÜNER AND DANIEL DRUCKMAN
Equilibrium
The solution for incomplete-information games in strategic form is
the Bayesian equilibrium. It is the counterpart of the Nash equilibrium for complete-information games.7 Players choose strategies
that maximize their expected payoffs given their updated beliefs
about each other. These strategies are also best replies against each
other. Since one player has two types and the other only one, the
posteriors do not differ from the prior beliefs.8
A Bayesian equilibrium concept is not needed to solve this game.
The dominant-strategy equilibrium suffices to give the solution: both
players choose their dominant strategies; they eliminate their dominated choices, and this holds for any misperception of Nibmuarea.
Neither of the Kadashman-Enlil types has an incentive to switch
from his strategy since, by choosing his dominant strategy, Nibmuarea shows the princess. The unique equilibrium is {S, (D, −D);
for all p, 0 ≤ p ≤ 1}. This is again an intersection between the
dominant strategies.
Interpretation
The Pharaoh has the dominant strategy of showing the princess against both types of Kadashman-Enlil. The equilibrium does
not ascribe two different actions for Nibmuarea depending on
Kadashman-Enlil’s type. Kadashman-Enlil also does not choose his
dominated strategy depending on his type. Hence, the Bayesian
equilibrium is similar to the equilibrium computed in the completeinformation game.
In this equilibrium, there is the possibility that Kadashman-Enlil
is indeed interested in his sister’s fate and chooses to send dignitaries. Nibmuarea takes this intention into account in eliminating
his dominated strategy: In the previous game the elimination was
done solely against the Reluctant. Since Nibmuarea’s misperception is common knowledge, Kadashman-Enlil cannot expect that
Nibmuarea, who misperceives him, will show someone else as the
princess. Nibmuarea could not expect that Kadashman-Enlil could
expect that he will choose to show someone else because he knows
that Kadashman-Enlil knows that he (Nibmuarea) misperceives him.
Thus, Nibmuarea’s misperception would not lead Kadashman-Enlil
IDENTIFICATION OF A PRINCESS UNDER INCOMPLETE INFORMATION
393
to think that there is a chance that Nibmuarea will present someone
else as the princess to his messengers.
The dominant-strategy equilibrium of the previous game and
the equilibrium computed here give two different rationales for
the observed outcome according to Nibmuarea. In the former, the
(S, −D) outcome is due to the dominant strategies of both sides,
with Pharaoh being informed of Kadashman-Enlil’s preferences.
In the latter, Nibmuarea again shows the princess and the sending
of non-dignitaries indicates that Kadashman-Enlil is the Reluctant.
Kadashman-Enlil maintains that (−S, −D) may be the outcome.
However, if we assume Nibmuarea is genuine in his words, there
is no support for Nibmuarea’s presentation of someone else to the
non-dignitaries. There is no possibility for the outcome (−S, −D)
to be reached in any equilibrium of this game. Nevertheless, one
may suspect that Nibmuarea was not genuine and perhaps prone to
present someone else as the princess. We now turn to that aspect of
the interaction.
Kadashman-Enlil’s misperception
Kadashman-Enlil could possibly have misperceived Nibmuarea.
When he is the Reluctant, he could have believed Nibmuarea had
an incentive to present someone else as his sister to the messengers.
In this version of the game two types represent Nibmuarea: one has
a genuine interest in presenting the princess as he affirms in EA1.
He is called the “Genuine”. The other prefers to show someone else
to non-dignitaries but shows the princess to the dignitaries. He is
called the “Feigned”. The first case occurs with probability q and
the second one with (1 − q). These probabilities are KadashmanEnlil’s prior assessments about Pharaoh’s type. These probabilities
are common knowledge, and Nibmuarea knows his own preference
ordering.
In the first matrix of Figure 3, Kadashman-Enlil thinks the
Pharaoh is genuine in his words. Nibmuarea shows the princess
either to non-dignitaries or dignitaries, his dominant strategy. In the
second matrix of the figure, the Feigned suffers a cost of −h, where
h > 0, when the princess is presented to non-dignitaries. He therefore has no dominant strategy. Showing the princess to dignitaries
is the best strategy. Showing someone else is best if non-dignitaries
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SERDAR GÜNER AND DANIEL DRUCKMAN
Figure 3. Identification under Kadashman-Enlil’s misperception.
are sent. This condition adds another dimension to EA1: it contradicts the words of the Pharaoh but incorporates Kadashman-Enlil’s
suspicion.
Equilibrium
The Genuine has a dominant strategy of showing the princess (S)
while the Feigned has no such strategy. Thus, we will never have
(−S, S) and (−S, −S) but either (S, S) or (S, −S). The first typecontingent strategy is: “I show the princess whether I am genuine or
not” (S, S); the latter is: “I show the princess if I am the Genuine,
and do not if I am the Feigned” (S, −S). Kadashman-Enlil has only
one type, so his strategies are D and −D. Similar to the previous
analysis, Kadashman-Enlil has a dominant strategy of −D, and he
will use it regardless of Nibmuarea’s strategies (S, S) and (S, −S).
There are two equilibrium candidates: {−D, (S, S); for all q, 0 ≤
q ≤ 1} and {−D, (S, −S); for all q, 0 ≤ q ≤ 1}. However, since
Kadashman-Enlil chooses −D against (S, S), the Feigned has an
interest in deviating from showing the princess to showing someone
else. So, {−D, (S, S); for all q, 0 ≤ q ≤ 1} is not a Bayesian
equilibrium. The unique Bayesian equilibrium is −D, (S, −S); for
all q, 0 ≤ q ≤ 1}. No player has an incentive to switch unilaterally
from his equilibrium strategy when the other sticks to his.
IDENTIFICATION OF A PRINCESS UNDER INCOMPLETE INFORMATION
395
Interpretation
Kadashman-Enlil should not send dignitaries due to his misperception of the Pharaoh. Similar to Nibmuarea’s behavior in the previous
game, either if q = 0 or q = 1 or for any q between 0 and 1, he
has a dominant strategy of sending non-dignitaries. Nibmuarea’s
behavior, given Kadashman-Enlil’s misperception, is more complex
than Kadashman-Enlil’s behavior given Nibmuarea’s misperception: the Feigned would not find it beneficial to hide his preferences.
Nibmuarea, knowing Kadashman-Enlil’s uncertainty and dominant
strategy of sending non-dignitaries as messengers, would not show
the princess even if he would like to fool the Babylonian ruler by
showing the princess to the non-dignitaries. Once Kadashman-Enlil
is assumed to have a dominant strategy of sending non-dignitaries,
the outcomes of the game should either be the presentation of the
princess or the presentation of someone else to the non-dignitaries.
Thus, while the Feigned has no dominant action, he reveals his
preferences.
These games of incomplete information enrich the completeinformation analysis by introducing beliefs. Equilibrium strategies
remain the dominant ones. Both anticipate that the Genuine always
shows the princess while the Feigned does not, and the Keen sends
dignitaries while the Reluctant sends non-dignitaries. So, there is
game-theoretic support for the observed outcome of presenting the
princess to non-dignitaries. Pharaoh might have indeed been the
Genuine and Kadashman-Enlil the Reluctant with the aim of aggrandizing himself. It may also be, as Kadashman-Enlil contends,
that Nibmuarea was the Feigned and presented someone else to the
non-dignitaries.
We could believe in Nibmuarea’s words if Nibmuarea had the
dominant strategy of showing the princess under both complete and
incomplete information conditions, and, Kadashman-Enlil’s suspicions are understood if Kadashman-Enlil indeed misperceived
Nibmuarea. Yet, both rulers might have misperceived each other.
Such compound misperceptions approximate the interaction better
but the analysis becomes more complex.
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A TWO-SIDED INCOMPLETE-INFORMATION GAME
The games matrices shown below in Figures 4 and 5 represent an
interaction in which both Nibmuarea and Kadashman-Enlil misperceive each other. The probabilities p, q, r, and s are nature’s choices
of a particular type combination: they are respectively players’ prior
assessments about nature’s choice of type combinations (G, K), (G,
R), (F, K), and (F, R), where G denotes the Genuine, F the Feigned,
K the Keen and R the Reluctant. The sum of these probabilities is
equal to one. The four type combinations are mutually exclusive and
exhaustive.
The first cell in Figure 5 represents an interaction between
the Genuine and the Keen. The second represents an interaction
between the Genuine and the Reluctant. The third cell represents
an interaction between the Feigned and the Keen, and the last one
is an interaction between the Feigned and the Reluctant. The figure
is useful in distinguishing between these combinations and introducing a notation for the priors in a two-sided incomplete-information
game.
Players are assumed to be informed about the possible types of
each other and the associated probability distribution p, q, r, s. We
further assume that these types and the associated probability distribution are common knowledge. Nibmuarea and Kadashman-Enlil’s
payoffs depend on the strategy combinations and their own types.
Given their types, Nibmuarea and Kadashman-Enlil update their
priors. Here, unlike in the one-sided incomplete-information games,
updating matters. For example, given that he is the Genuine,
Nibmuarea’s posterior that Kadashman-Enlil is the Keen is:
µ′ni (K|G) = µ(G, K)/[µ(G, K) + µ(G, R)] = p/(p + q).
We similarly calculate Nibmuarea’s posteriors for each of the other
combinations of intentions:
µ′ni (R|G) = µ(G, R)/[µ(G, R) + µ(G, K)] = q/(q + p).
µ′ni (K|F) = µ(F, K)/[µ(F, K) + µ(F, R)] = r/(r + s).
µ′ni (R|F) = µ(F, R)/[µ(F, R) + µ(F, K)] = s/(s + r).
IDENTIFICATION OF A PRINCESS UNDER INCOMPLETE INFORMATION
Figure 4. Identification under two-sided misperception.
Kadashman-Enlil’s four posteriors are:
µ′ke (G|K) = µ(G, K)/[µ(G, K) + µ(F, K)] = p/(p + r).
µ′ke (F|K) = µ(F, K)/[µ(F, K) + µ(G, K)] = r/(r + p).
µ′ke (G|R) = µ(G, R)/[µ(G, R) + µ(F, R)] = q/(q + s).
µ′ke (F|R) = µ(F, R)/[µ(F, R) + µ(G, R)] = s/(s + q).
397
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SERDAR GÜNER AND DANIEL DRUCKMAN
Figure 5. Type combinations and priors.
Nibmuarea has four type-contingent strategies: (S, S), (S, −S),
(−S, S), and (−S, −S). The first letter indicates the Genuine’s
strategy choice, the second the Feigned’s. Kadashman-Enlil’s typecontingent strategies are similarly (D, D), (D, −D), (−D, D), and
(−D, −D). Again, the first letter refers to the Keen’s strategy choice,
the second one to the Reluctant. Nibmuarea and KadashmanEnlil maximize their expected payoffs by choosing an appropriate
strategy. The combinations of players’ type-contingent strategies
make up a 4 × 4 matrix of expected payoffs for each player.
Some strategies will never be chosen, so we handle a simpler
problem. The Keen has a dominant strategy of D and the Reluctant
a dominant strategy of −D. Thus we will only observe (D, −D) in
equilibrium. The choices (D, D), (−D, D), and (−D, −D) will not
be made. Similarly, the Genuine will never play −S, but the Feigned
has no dominant strategy; therefore we will not examine the strategy
choices of (−S, S) and (−S, −S) but only (S, S) and (S, −S).
Nibmuarea’s expected payoff from (S, S) against (D, −D) is:
Pni [(S, D), G, K]µ′ni (K|G) + Pni [(S, −D), G, R]µ′ni (R|G)
+ Pni [(S, D), F, K]µ′ni (K|F)
+ Pni [(S, −D), F, R]µ′ni (F|R)
= 0[p/(p + q)] + (−d)[q/(q + p)] + 0[r/(r + s)]
+ (−d − h)[s/(s + r)]
= −d[q/(q + p)] − (d + h)[s/(s + r)].
IDENTIFICATION OF A PRINCESS UNDER INCOMPLETE INFORMATION
399
Figure 6. Nibmuarea’s expected payoff matrix.
Nibmuarea’s expected payoff from (S, −S) against (D, −D) is:
Pni [(S, D), G, K]µ′ni (K|G) + Pni [(S, −D), G, R]µ′ni (R|G)
+ Pni [(−S, D), F, K]µ′ni (K|R)
+ Pni [(−S, −D), F, R]µ′ni (F|R)
= 0[p/(p + q)] + (−d)[q/(q + p)] + (−d − c)[r/(r + s)]
+ (−d)[s/(s + r)] = −d[1 + q/(q + p)] − c[r/(r + s)].
The expected payoff matrix for Nibmuarea is summarized in
Figure 6.
Kadashman-Enlil’s expected payoff from (D, −D) against (S, S)
is:
Pke [(D, S), G, K]µ′ke (G|K) + Pke [(−D, S), G, R]µ′ke (G|R)
+ Pke [(D, S), F, K]µ′ke (F|K)
+ Pke [(−D, S), F, R]µ′ke (F|R)
= 0[p/(p + r)] + (d)[q/(q + s)] + (0)[r/(r + p)]
+ (d)[s/(s + q)] = d.
Kadashman-Enlil’s expected payoff from (D, −D) against (S,
−S) is:
Pke [(D, S), G, K]µ′ke (G|K) + Pke [(−D, S), G, R]µ′ke (G|R)
+ Pke [(D, −S), F, K]µ′ke (F|K)
+ Pke [(−D, −S), F, R]µ′ke (F|R)
= 0[p/(p + r)] + (d)[q/(q + s)] + (d − l)[r/(r + p)]
+ (d)[s/(s + q)] = d + (d − l)[r/(r + p)].
Kadashman-Enlil’s expected payoff matrix, then, is shown in
Figure 7.
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SERDAR GÜNER AND DANIEL DRUCKMAN
Figure 7. Kadashman-Enlil’s expeced payoff matrix.
For Nibmuarea we have the following best-reply conditions:
Condition 1: if −d[q/(q + p)] − (d + h)[s/(s + r)] > – d [1 + q / (q
+ p)] – c [r / (r + s)], then (S, S).
Condition 2: if −d[q/(q + p)] − (d + h)[s/(s + r)] < −d[1 + q/(q +
p)] − c[r/(r + s)], then (S, −S).
Condition 3: if −d[q/(q + p)] − (d + h)[s/(s + r)] = −d[1 + q/(q +
p)] − c[r/(r + s)], then, we are indifferent between (S, S) and (S,
−S).
The type-contingent strategy (S, S) means that Nibmuarea shows
the princess whether he is genuine or not (facing either the Keen or
the Reluctant); in all four matrices Nibmuarea chooses S. Suppose
that condition 1 is satisfied, s/r < (d + c)/h. Nibmuarea chooses (S,
S) against (D, −D). Against this choice, Kadashman-Enlil has no
incentive to deviate from (D, −D). The first Bayesian equilibrium is
{(S, S), (D, −D); s / r < (d + c) / h}.
Suppose now that the second condition is satisfied. So, we have
s/r > (d + c)/h. Nibmuarea’s type-contingent best reply against
(D, −D) is (S, −S). Hence, in the first and second matrices Nibmuarea chooses S, and in the third and fourth matrices he chooses
−S. Against this choice, Kadashman-Enlil again has no incentive
to deviate from (D, −D). The second Bayesian equilibrium is {(S,
−S), (D, −D); s / r > (d + c) / h}.
If the third condition is satisfied, then Nibmuarea can choose
either S in all four matrices or S in the first and second matrices
and −S in the third and fourth ones. Kadashman-Enlil who has the
unique type-contingent strategy made up of his dominant strategies
should not deviate from (D, −D). So, such a mixture of (S, S) and
(S, −S) does not lead to Kadashman-Enlil’s indifference. Therefore,
the third Bayesian equilibrium is: {(S, −S) or (S, S), (D, −D); s / r
= (d + c) / h}.
IDENTIFICATION OF A PRINCESS UNDER INCOMPLETE INFORMATION
401
Interpretation
If Nibmuarea cheats and messengers detect the cheating, his reputation as the greatest among Middle-Eastern rulers of that time would
be severely tarnished: Why should a Pharaoh double-cross the messengers sent by the ruler of a lesser kingdom? If he cheats, he is also
sure that Kadashman-Enlil will break the Egypt-Babylon dynastic
alliance and will deny his daughter to Nibmuarea. The Babylonian
alliance is not as important as is the loss of his reputation. This cost
of reputation loss is higher or at least equal to the cost of being
denied a daughter by the Babylonian ruler. We assume that c > d.
Nibmuarea suffers the cost h when he prefers to present the princess
to dignitaries. When the princess is presented to non-dignitaries,
Kadashman-Enlil can only pretend, but cannot be sure, that the
Pharaoh is cheating, thus denying his daughter to Nibmuarea. This
cost is less than the cost of reputation loss. Thus, we also assume
that c > h.9
Therefore, given that the Keen sends dignitaries and the Reluctant sends non-dignitaries, both types of Nibmuarea (the Genuine
and the Feigned) find the presentation of the princess as being more
beneficial than the strategy “present the princess if I am genuine and
do not if I am feigned.” The Feigned’s cost of cheating is greater than
any cost suffered from presenting the princess to non-dignitaries. If
the Feigned assesses that it is more probable that he will face the
Keen than the Reluctant, he presents the princess. The presentation
of someone else gives a smaller payoff.
Even if s > r, c is sufficiently greater than h, the condition of
presenting the princess is still satisfied. Thus, while the Feigned
judges that there is a greater probability that he faces the Reluctant,
he still chooses to present the princess since the cost of cheating is
quite high. Both Nibmuarea types then choose the same strategy of
presenting the princess under these conditions.
This equilibrium implies that Kadashman-Enlil’s claim is not
grounded: the one presented to the messengers was indeed the princess. This holds true even if he perceived Nibmuarea as the Feigned.
For the same reason, the first equilibrium provides support for Nibmuarea’s words in EA1. Nibmuarea should indeed have no reason
to conceal the death of the princess and to present someone else,
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SERDAR GÜNER AND DANIEL DRUCKMAN
because the loss of reputation loomed large whether the Keen sends
dignitaries or the Reluctant sends non-dignitaries.
The second condition gives d[s/(s + r)] + h[s/(s + r)] > d +
c[r/(r + s)]. This implies that h s > r(d + c). As c > h, and d > 0,
if r > s, the condition cannot be satisfied. Given that the Keen sends
dignitaries and the Reluctant sends non-dignitaries, the Genuine
presents the princess. The Feigned also does not find that his best
reply is the presentation of someone else. This holds if the Feigned
thinks that it is equally probable that he is likely to face the Keen
or the Reluctant, or that it is more probable that Kadashman-Enlil is
the Keen.
This result contrasts with the equilibrium of the one-sided
incomplete-information game that models Kadashman-Enlil’s misperception. In that game, the Feigned shows someone else to nondignitaries even though he can suffer the cost of reputation loss,
because he knows that Kadashman-Enlil is the Reluctant. In the
first equilibrium, however, the Feigned prefers to show the princess
because there is a chance of facing the Keen.
Against Kadashman-Enlil’s type-contingent strategy, the Feigned
prefers to show someone else if he assesses that he is more likely to
face the Reluctant than the Keen, that is, if s > r. Under such a
strict inequality where r is sufficiently small there is a chance for
the second condition to hold. Therefore, Nibmuarea’s types take
different actions if it is more likely that the game is played between
the Feigned and the Reluctant rather than between the Feigned and
the Keen.
This rationale is similar to the equilibrium computed in the
one-sided incomplete-information game which models KadashmanEnlil’s misperception. Given that the Reluctant Kadashman-Enlil
sends non-dignitaries, the Feigned Nibmuarea prefers to show
someone else. Hence, Nibmuarea shows the sister if he perceives
that Kadashman-Enlil is sufficiently keen to learn of his sister’s fate.
Assuming that Nibmuarea indeed showed the princess, he must have
thought that Kadashman-Enlil was the Keen and was surprised to
see non-dignitaries at his Court.
The third condition implies h s = r (d + c). The indifference
means that Nibmuarea can show the princess regardless of his
and Kadashman-Enlil’s types. Since the Genuine shows the prin-
IDENTIFICATION OF A PRINCESS UNDER INCOMPLETE INFORMATION
403
Figure 8. Location of equilibria under two-sided misperception.
cess anyway, this condition implies that the Feigned is indifferent
between showing the princess or someone else. Again, if s = r
or s < r, the condition cannot be satisfied. Given KadashmanEnlil’s type-contingent strategy, the Feigned cannot be indifferent
between presenting the princess or someone else if he thinks that
Kadashman-Enlil can be either the Keen or the Reluctant, or that
Kadashman-Enlil is in fact interested in his sister’s fate (the Keen).
Geometric interpretation
All three equilibria can be presented on the unit square taking the
parameters h, d, and c as constants and the priors r and s as variables
with values between 0 and 1. Condition 3 is the equation of a line,
s = [ (d + c) / h] r, passing through the origin with a positive slope
of (d + c) / h. The slope is greater than 1 as d + c > h under the
payoff assumptions. The line in Figure 8 divides the unit square into
two regions. The unit square contains all pairs of (r, s), that is, priors
indicating the interaction between the Feigned and the Keen, and
the one between the Feigned and the Reluctant respectively. The
diagonal line contains all pairs (r, s) satisfying the equality s h = r (d
+ c).
Hence, the pairs of (r, s) leading to Nibmuarea’s indifference
satisfy the equation of the line s = [ (d + c) / h] r. The line gives
the third equilibrium. In the triangular region A, the condition s h >
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SERDAR GÜNER AND DANIEL DRUCKMAN
r (d + c) holds; likewise the condition s h < r (d + c) holds in region
B. Thus, in A, Nibmuarea’s best reply is to “show the princess if
I am the Genuine and someone else if I am the Feigned”, and this
gives the second equilibrium. The first equilibrium is given by B
where Nibmuarea’s best reply is “show the princess whether I am
the Genuine or the Feigned”.
The area of A is h / 2 (d + c), and the area of B is equal to 1–h
/ 2 (d + c). Since c and d increase while h is constant, A shrinks.
The slope of the line becomes steeper and the line approaches the
ordinate. Consequently, r diminishes while s remains constant in A.
In contrast, A gets larger if h increases while c and d are constant.
This implies that if both the cost of reputation loss and the value
of the dynastic marriage increase, a smaller prior that the Feigned
faces the Keen for the same value of s can satisfy the condition s h >
r (d + c). For the Feigned to present someone else and the Genuine to
present the princess, a smaller prior that the Feigned faces the Keen
suffices. This is consistent with the previous observation that against
Kadashman-Enlil’s type-contingent strategy, the Feigned prefers to
show someone else if he thinks that he faces the Reluctant rather
than the Keen, that is, if s > r.
In contrast to the previous result, if the Feigned’s cost h increases while c and d are constant, then A gets larger. If the Feigned
suffers a relatively greater cost when presenting the princess to nondignitaries, then a pair (r, s) previously not in A can be included in
A. A greater prior that the Feigned faces the Keen for the same value
of s can satisfy the condition s h > r (d + c). Hence, if the costs of
presenting the princess to non-dignitaries are sufficiently high, then
even if the Feigned thinks that he will eventually face the Keen, he
presents someone else.
CONCLUSION
These equilibria indicate that if Nibmuarea was the Feigned he must
have thought that Kadashman-Enlil was interested in his sister’s
fate, since he presented the princess even to the non-dignitaries. But,
if as the Feigned he thought that Kadashman-Enlil was primarily interested in aggrandizing himself, he would have presented someone
else to the non-dignitaries. Our analysis of the complete-information
IDENTIFICATION OF A PRINCESS UNDER INCOMPLETE INFORMATION
405
case supports Nibmuarea’s statements in EA1. The same applies
in the case of one-sided incomplete-information: the princess is
always presented by Nibmuarea under that condition. In the twosided incomplete-information game, both types of Nibmuarea show
the princess. If the Feigned thinks that Kadashman-Enlil is in fact
interested in his sister’s fate, the cost of cheating is quite high. Nevertheless, Kadashman-Enlil’s suspicions that Nibmuarea preferred
to show the princess only to non-dignitaries would prove correct
if Nibmuarea, in his role as the Feigned, attributed self-centered,
aggrandizing motives to his opposite number.
This game-theoretic analysis of rational choices under different
information conditions reveals the interactive reasoning that may
have taken place between the two rulers. Choice of strategy is
contingent on the player’s beliefs about the other’s intentions. Of
course, we do not have the possibility of confirming these mutual expectations, suspicions, and choices. Nevertheless, both rulers’
claims find support in these equilibria. Consequently, strategic
uncertainty characterized the interactions depicted in EA1.
The analysis of strategic uncertainty is an important contribution
of game theory. Alternative approaches to the analysis of history
provide limited insights into the implications of such uncertainty
including the consequences of unrevealed intentions and suspicions,
as well as the implementation of interactive choices. We have,
in this article, reconstructed a bronze-age story in terms of contemporary strategic thinking. Our reconstruction demonstrated the
power of “what . . . if” reasoning: Plausible outcomes were derived from assumptions about available choices as revealed in the
Amarna letters. We assumed first that the Pharaoh’s account of the
situation was correct, namely that he had a dominant strategy and
that Kadashman-Enlil did not send non-dignitaries. We then relaxed these assumptions, one-by-one, in the incomplete-information
games, and discovered that the Pharaoh’s account may not have been
correct. The Feigned Pharaoh could well have presented someone
else to the dignitaries sent by the Keen king. On the other hand,
when d (marriage value) and c (cost of cheating) increase, it is
more likely that both types make the same choice: present the princess. This analysis is particularly significant in light of missing
information in other letters about the sequel to these interactions.
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SERDAR GÜNER AND DANIEL DRUCKMAN
NOTES
1. Ankhesenamun, Egyptian queen and daughter of the Pharaoh Amenophis IV
was a childless widow; thus, she asked the powerful Hittite king Suppiluliumas to send her one of his sons to rule Egypt (Gurney,1990: 24–25).
2. Italics in original. To put oil on the head of a girl means a step towards
marriage in the Bronze Age (like an engagement).
3. Moran (1992: xviii) indicates the exceptional quality of the EA1: it is a copy of
the translation from Egyptian into Akkadian and kept in the foreign language
section of the Egyptian archives. As a rule, the Egyptian versions are filed in
the archives and the translated letters are sent (Moran, xvii).
4. Kaska, a region in Anatolia’s northwest Black Sea coast, was the land of
ferocious tribes. They regularly attacked Hittites of Anatolia, the competitor
of Egypt during the Middle-Bronze Age. Hanigalbat was a Mitanni kingdom
based in upper-eastern Mesopotamia, a part of southeast Anatolia. Ugarit was
a city-kingdom on the Syrian coast under the influence of Egypt.
5. Showing the princess dominates showing someone else if showing her is as
good as not showing her to non-dignitaries and better than not showing her if
dignitaries are sent.
6. For convenience, row players are chosen to represent those who have incomplete information.
7. Precisely, Bayesian equilibrium is a strategy choice that maximizes players’ expected payoffs against others’ equilibrium strategies. Under the belief
conditions, players gain nothing or suffer utility losses when they deviate unilaterally from their equilibrium strategies given that others stick to
their strategies. For illustrative examples see Eichberger 1993: 123–151 and
Friedman 1986: 48–67.
8. Since we have one type for the Pharaoh but two types for Kadashman-Enlil,
there are two type combinations (T, K) and (T, R) where T denotes Pharaoh’s
type, K the Keen, and R the Reluctant. The probabilities p and (1 − p)
respectively represent the selection of (T, K) and (T, R), the selection of a particular type-pair. Nibmuarea updates his priors about Kadashman-Enlil’s type
given his type is T. These updated beliefs are Nibmuarea’s posterior beliefs
about Kadashman-Enlil. Pharaoh’s posterior about Kadashman–Enlil of type
K given that he is of type T is calculated with the help of the Bayes’ theorem:
µ(K|T) = µ(T, K)/(µ(T, K) + µ(T, R)) = p/(p + 1 − p) = p. Similarly,
Pharaoh’s posterior about Kadashman-Enlil of type R given that he is of type
T is: µ(R|T) = µ(T, R)/(µ(T, R) + µ(T, K)) = (1 − p)/(1 − p + p) = 1 − p.
9. The first condition implies (d + h)[s/(s + r)] < d + c[r/(r + s)]. This gives
d[s/(s + r) + h[s/(s + r)] < d + c[r/(r + s)]. So, we have (d + c)r > h s. As
0 < s/(s + r) < 1, we have d[s/(s + r)] < d, for the inequality to be satisfied
it suffices to have h[s/(s + r)] ≤ c [r/(r + s)]. The inequality in turn implies
h s ≤ c r. Under the assumption of c > h, the condition h s ≤ c r is satisfied
if r ≥ s.
IDENTIFICATION OF A PRINCESS UNDER INCOMPLETE INFORMATION
407
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Address for correspondence: Daniel Druckman, Institute for Conflict Analysis
and Resolution, George Mason University, 4400 University Drive, Fairfax, VA
22030-4444, USA
Phone: (703)993-3655; Fax: (703)993-1302; E-mail: ddruckma@gmu.edu